Extent of hydrogen coverage of Si(001) under chemical ... · thermodynamic equilibrium was studied...

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Extent of hydrogen coverage of Si(001) under chemical vapor

deposition conditions from ab initio approaches

Phil Rosenow, Ralf Tonnera)

Fachbereich Chemie and Wissenschaftliches Zentrum für Materialwissenschaften, Philipps-Universität Marburg, Hans-

Meerwein-Straße, 35032 Marburg, Germany

The extent of hydrogen coverage of the Si(001)c(4x2) surface in the presence of hydrogen gas has

been studied with dispersion corrected density functional theory. Electronic energy contributions

are well described using a hybrid functional. The temperature dependence of the coverage in

thermodynamic equilibrium was studied computing the phonon spectrum in a supercell approach.

As an approximation to these demanding computations, an interpolated phonon approach was

found to give comparable accuracy. The simpler ab initio thermodynamic approach is not accurate

enough for the system studied, even if corrections by the Einstein model for surface vibrations are

considered. The on-set of H2 desorption from the fully hydrogenated surface is predicted to occur

at temperatures around 750 K. Strong changes in hydrogen coverage are found between 1000 and

1200 K in good agreement with previous reflectance anisotropy spectroscopy experiments. These

findings allow a rational choice for the surface state in the computational treatment of chemical

reactions under typical metal organic vapor phase epitaxy conditions on Si(001).

Keywords

Density functional theory; metal organic vapor phase epitaxy; semiconductor; ab initio

thermodynamics; silicon; surface state

a) Author to whom correspondance should be adressed. Electronic mail: [email protected].

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I. INTRODUCTION

The functionalization of silicon surfaces with organic or inorganic molecules extends the

range of applications for this important semiconductor material. One promising field is “silicon

photonics”, i.e. the construction of optically active devices based on silicon substrates.1 The most

important aim in this field is building a laser, for example by monolithically integrating an optically

active layer on silicon substrate. Promising candidates here are quantum well structures with group

13 and group 15 elements (aka. III/V), for example the material system Ga(N, As, P).2 These

materials are usually produced by metal organic vapor phase epitaxy (MOVPE), a method that

deposits liquid-state precursor molecules on heated surfaces via chemical vapor deposition.3 The

structure of the final thin-film and thus the properties of the resulting device are crucially

determined by the chemical reactions in the nucleation phase of the deposition process.4 This

guides our ongoing research endeavors toward the chemical reactivity in the MOVPE process by

quantum chemical methods.5 Our intention to extend these investigations to surface reactivity

requires an in-depth knowledge about the state of the surface under MOVPE conditions, which is

the topic of this study.

The semiconductor surface most often used in MOVPE experiments is Si(001) due to its

technological importance.3 This surface undergoes the well-known dimer reconstruction and

buckling leading to Si(001)c(4x2) being the pristine state (Scheme 1f).6 The silicon atoms at the

surface exhibit “dangling bonds”, i.e. unpaired electrons which explains the high reactivity towards

adsorbates.7 Therefore, it is very likely that the commonly used carrier gas H2 will react with the

surface atoms under the elevated temperatures used for the deposition process. The final state for

hydrogenation under most experimental conditions is one hydrogen atom per surface dimer atom

(Scheme 1a), which is defined as one monolayer coverage ( = 1 ML) is this study. The

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monohydride termination of the Si(001) surface under MOVPE conditions was confirmed with

infrared spectroscopy.8 The temperature dependence of this hydrogen coverage is an important

factor for correct modelling of the growth reactions and is not well documented up to now.

Experimental surface science studies are mostly performed under ultra-high vacuum (UHV)

conditions and have given valuable insight in the dynamics of hydrogen adsorption, desorption

and diffusion on the surface in the past.9,10,11 Likewise, computational studies have been devoted

to desorption mechanisms and surface dynamics.12,13,14 Two studies using temperature

programmed desorption (TPD) in UHV conditions show an onset of H2-desorption from

hydrogenated Si(001) surfaces to occur between 700-780 K15 and 750±25 K16, respectively.

Brückner et al. studied the desorption of H2 from hydrogenated Si(001) both in a flow of hydrogen

and nitrogen carrier gas with reflectance anisotropy spectroscopy (RAS) under MOVPE

conditions.17 In a nitrogen flow, hydrogen desorbed from the H/Si(001) surface (Scheme 1a) at

750 K within 30 minutes leaving a pristine Si(001) surface in accordance with the TPD results

cited above. Under typical MOVPE conditions with hydrogen carrier gas, dehydrogenation

gradually increased with increasing temperatures. Lowering the temperature from 1270 K to 670

K, a strong change in the RAS signal was observed around T = 1070 K, which is associated with

significant hydrogenation of the bare surface. At lower temperatures, the signal of the fully

hydrogenated surface is found when compensating for a thermal drift in the measurements. The

authors thus concluded that the silicon surface is hydrogenated for temperatures below 1070 K in

a H2-atmosphere and pristine at 1270 K. Extrapolation from the low pressures used in the study to

atmospheric pressure of hydrogen yields = 0.94 ML coverage at 1070 K and = 0.25 ML

coverage at 1270 K. It should be stressed, that we will focus solely on surface coverage in

thermodynamic equilibrium – adsorption and desorption dynamics are not aimed at here and were

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extensively reviewed elsewhere.9-14 Likewise, the H2 induced formation of reconstruction

domains18 is not the aim of this study.

The theoretical treatment of thermodynamic properties of extended systems requires

performing phonon calculations for large supercells.19 These are computationally expensive,

especially for structures with low symmetry as for example submonolayer coverages (Scheme 1c,

d, e) and a more efficient approach is desirable. Such an alternative can be found in the ab initio

thermodynamics approach as recently reviewed by Reuter.20,21,22 Here, thermodynamic

contributions from the surface are neglected in a first approximation. In order to validate this

approach, careful comparison with full phonon calculations and experiment will be carried out.

We will focus on the temperature range between T = 300 and 1500 K, typical for growth

experiments.23

In section 2 we describe the computational methods employed. In section 3 we present the

results from ab initio thermodynamics (AITD), explicit phonon (EP) and interpolated phonon (IP)

computations, leading to a description of coverage dependence on temperature, which we compare

to the experimental results described above. We aim at finding the most efficient computational

approach to describe the extend of hydrogen coverage of Si(001) under MOVPE conditions.

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II. METHODS

A Computational details

Electronic structure calculations based on density functional theory (DFT) were carried out

with the Vienna ab-initio simulation package (VASP)24 version 5.3.5. Structures were optimized

using the PBE (Perdew-Becke-Ernzerhof) functional25 based on the generalized gradient

approximation and adding a semiempirical dispersion correction scheme by Grimme et al. (PBE-

D3(BJ))26. For the D3-scheme, the cutoff radius used is 50.2 Å, and the radius to determine the

coordination number was set to 21.2 Å. Additionally, the hybrid functional HSE06-D3(BJ)27 was

used with D3 parameters of the PBE0 functional. The D3 parameters are summarized in Table

S128 and are taken from Ref. 26b. A plane wave basis set combined with the projector augmented

wave method29 and an energy cutoff of 350 eV was used. The reciprocal space was sampled using

a Γ-centered Monkhorst-Pack k-mesh30 with a (4 2 1)-division for the Si(001) surface for a

supercell containing four dimers (Scheme 1).

SCHEME 1. Supercells of the Si(001) surface with different coverages θ used in this study.

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Electronic energies were converged to 10-5 eV while forces for optimizations were

converged to 10-2 eV/Å. For phonon calculations, the Phonopy 1.8.2 code31,32 was used to generate

supercells and compute thermodynamic properties from forces computed with VASP. Throughout,

asymmetric slabs with eight layers of silicon atoms were used (six and ten layers for convergence

study), where the two bottom layers where kept fixed at bulk positions and saturated with hydogen

atoms (frozen double layer approximation). For phonon calculations, (2x2) supercells ((3x3) cells

for convergence study) containing 16 dimers were used. A Γ-centered q-mesh with (8 4 1)-division

was applied in the determination of the phonon density of states. The theoretically optimized lattice

constant of Si was used (a = 5.421 Å; exp.: 5.415 Å)33. All thermodynamic correction terms were

derived with the computationally more efficient PBE-D3 functional.

The AITD approach used here follows the one described in Ref. 20. While the formulae

are derived to give Helmholtz energies, we neglect the volume change of the solid surface and will

report Gibbs Energies (G) throughout this manuscript. Thus, the Gibbs Energy change upon

dehydrogenation is defined as

∆ ∆ ∆ ln°

. (1)

Here, is the number of desorbing hydrogen atoms per unit cell, is the surface area of the unit

cell, is the universal gas constant, is the temperature, is the hydrogen partial pressure and °

is the standard pressure (1013 mbar); the electronic energy difference ∆ (2) is also called binding

energy

∆ , (2)

where is the energy of a cell with coverage , the energy of a hydrogen molecule, and

the energy of a cell with = 1 ML (see above). Finally, ∆ is the change in chemical

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potential. In the simplest approximation, it is the chemical potential required to remove one

hydrogen atom from a gas phase reservoir. These values are tabulated and the NIST (National

Institute of Standards and Technology) offers parameters that can be used with the Shomate

equations (3) to obtain accurate enthalpy and entropy values of hydrogen in the gas phase:34

°

, (3a)

ln , (3b)

where, . The parameters used in this work are given in Table I.

Another term arises since for any coverage 0 1 several possibilities for distributing

the adsorbate atoms on the adsorption sites in the cell exist. This leads to the configurational

entropy term

1 ln 1 ln , (4)

where is the number of possible adsorption sites.

In the EP approach, the Free Energy difference is computed analogous to the electronic

energy difference, but thermodynamic corrections from phonon calculations are added to the

electronic energies from DFT runs. These are computed via the supercell approach, where the

phonon modes are determined by dislocating atoms and computing the corresponding force

constants. The force constants are then further used to determine the phonon density of states

(DOS) and thermodynamic corrections.32 The thermodynamic correction term for hydrogen is

taken from the NIST.

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TABLE I. Parameters used for the calculation of µ(H2) with the Shomate equation (eq. (3)).[34]

298 –

1000 K 1000

– 2500 K

A 33.0662 18.5631

B -11.3634 12.2574

C 11.4328 -2.85979

D -2.77287 0.26824

E -0.15856 1.97799

F -9.9808 -1.14744

G 172.708 156.288

The temperature range for the thermodynamic investigations is 300 – 1500 K, subdivided in 10 K-

steps. For the assessment of the approaches outlined above, a typical pressure for MOVPE growth

has been used (p = 50 mbar). Furthermore, p was varied up to atmospheric pressure to elucidate

the influence of this parameter on the thermodynamic data.

Cartesian coordinates and total energies of all compounds discussed in the text are available in the supplementary

material.28

B Convergence studies

For the phonon computations, convergence with respect to slab thickness and supercell size

had to be ensured. The convergence of the phonon DOS was used as criterion (shown in Figure S1

in the supplemental material28). The phonon DOS for (2x2) and (3x3) supercells are very similar,

thus the smaller cell is sufficient. The peaks in the DOS associated with Si-H vibrations (> 85 cm-

1) are invariant to changes in the slab thickness from six to ten layers, while regions reflecting

lattice vibrations (< 85 cm-1) of the silicon slab change in intensity according to the increased

number of Si atoms for thicker slabs. Vibrations at low frequencies contribute significantly to the

entropy corrections. For the relative energies we report here, there will be a significant error

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compensation regarding this term. Thus, a slab with eight layers of Si atoms and (2x2) supercells

for the phonon calculations were used here.

III. RESULTS AND DISCUSSION

Desorption energies (E) for cleavage of H2 from fully hydrogen-covered H/Si(001)

(Scheme 1a, 1b) to sub-monolayer coverages = 0.75, 0.5, 0.25 ML (Scheme 1c-e) and the clean

surface ( = 0 ML, Scheme 1f) are given in Table 2 derived with the PBE-D3 and HSE06-D3

functionals according to the general desorption reaction (5):

Si H → Si H . (5)

We use the sign convention of positive E values indicating an energetically unfavorable reaction.

It is found that the reaction energies for the desorption reaction (5) are positive for all coverages

(Table II). The desorption energy depends slightly on the final coverage, with desorption from =

1 ML to = 0.75 ML being the most endo-energetic process. However, this is probably a cell-size

effect since the short distance between the translational images can lead

TABLE II. Desorption energies per H atom (E, see eq. (5)) from a monolayer to various coverages computed with PBE-D3 and HSE06-D3.

θfin 0.75 0.50 0.25 0.00 Eava

PBE-D3 92.9 88.8 90.1 88.7 90.1

HSE06-D3 108.2 104.2 105.4 104.0 105.5 aaveraged value for all coverages

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FIGURE 1. Ab initio thermodynamics (AITD) approach: GAITD values according to reaction (5)

based on HSE06-D3 energies at p = 50 mbar.

to spurious interactions due to the periodic boundary conditions employed. To eliminate this

unphysical effect, an average desorption energy is given in Table 2 which will be used for the rest

of the study. Desorption energies computed with HSE06-D3 are roughly 15 kJ mol-1 higher

compared to PBE-D3 results. The values computed with HSE06-D3 agree slightly better with a

range of other reported chemisorption energies,11 thus this functional is used further-on for E.

Thermodynamic correction of the electronic desorption energies was first carried out with

the AITD approach leading to Gibbs energies of desorption GAITD presented in Figure 1. The

fully covered surface is stable up to ca. 1025 K, = 0.75 ML is stable for a temperature range of

about 1025 - 1100 K, followed by a range of T = 1100-1280 K with = 0.5 ML. From 1280 –

1400 K, = 0.25 ML is the most stable range and the pristine surface is only found above ca. 1410

K. The broad temperature range for = 0.5 ML can be attributed to the configurational entropy

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FIGURE 2. Explicit phonon (EP) approach: GEP values according to reaction (5) based on

HSE06-D3 energies at p = 50 mbar.

term, which is largest for this coverage. Neglect of this term would lead to prediction of a complete

desorption of the adsorbed hydrogen atoms at T = 1040 K without intermediate coverages, which

is unphysical. Since the temperatures studied are rather high, the neglect of surface phonons is a

severe approximation and needs to be checked in the next step by taking into account

thermodynamic correction terms for the surface.

Deriving the phonon spectrum for the surface (EP approach) leads to the Gibbs energies of

desorption GEP presented in Figure 2. While the qualitative picture remains the same compared

to the AITD approach (Figure 1), the temperature for on-set of hydrogen desorption is reduced by

approximately 200 K and = 0.75 ML is found to be more stable than = 1 ML at T = 875 K.

This difference between EP and AITD approaches points toward a non-negligible thermodynamic

stabilization of the partially hydrogen-covered surface (Equation (5)). Notably, the temperature

range where = 0.25 ML is predicted to be stable is significantly narrower compared to the AITD

approach (ca. 60 K vs. 130 K for AITD). The pristine surface is reached at ca. 1240 K based on

the EP approach.

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Since deriving phonon spectra is computationally very demanding, especially for

intermediate coverage structures of low symmetry, a more feasible alternative is desirable. The

thermodynamic correction terms of intermediate coverage structures can be linearly interpolated

between the initial and final structures of full and no coverage:

∗ 1 ∗ . (6)

FIGURE 3. Interpolated phonon (IP) approach: GIP values according to reaction (5) based on

HSE06-D3 energies at p = 50 mbar.

FIGURE 4. Temperature dependence of coverage for IP and AITD approach. Binding energies

computed with PBE-D3 and HSE06-D3. The grey-shaded area indicates the range of the graph

with partially hydrogenated surface (0.95 > > 0.1 ML).

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Since there are eight possible adsorption positions for hydrogen atoms in the supercell, the amount

of hydrogen atoms transferred can be expressed as

1 8. (7)

The GIP values obtained with this IP approach are presented in Figure 3. The agreement between

the sophisticated EP approach and the much simpler IP ansatz is surprisingly good. Small

differences in the explicitly computed and interpolated chemical potential lead to different slopes

of the Gibbs energy and thus to slightly different temperatures of coverage change. But the

transition temperatures between the different coverages differ by only up to 40 K. However, these

differences can be neglected without changing conclusions.

For a real surface any intermediate coverage is possible, thus it is convenient to be able to

treat coverage as a continuous variable. Using the IP approach and the averaged electronic energy

difference, ΔGIP can be expressed in a form that leaves θ as the only remaining variable:

∆ ∆ ∆ , ln ° ln 1 ln 1 . (8)

For each temperature T, the coverage which minimizes GIP is the one which should be observed

experimentally. This leads to a particularly instructive representation shown in Figure 4. The

Figure here includes results from both functionals used in this study: PBE-D3 and HSE06-D3.

This (quasi) continuous formulation of temperature dependent coverage predicts a small initial

desorption which becomes larger for higher temperatures in line with the known desorption

dynamics of the system, e. g. visible in TPD experiments.15 The change of desorption with respect

to temperature is highest for coverages in the vicinity of = 0.5 ML. For high temperatures (T >

1400 K), the pristine surface is reached. A small residual coverage is predicted even for high

temperatures. It should be noted that the coverage trend for the IP approach based on HSE06-D3

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and the AITD approach based on PBE-D3 coincide to a large extent. Thus, there is obviously error

compensation in the description of the electronic and thermodynamic effects leading to the

apparent conclusion that the simpler functional (PBE-D3) with the simpler thermodynamic

approximation (AITD) gives accurate results.

In order to check if refining the AITD approach by applying the Einstein model35 to the

vibrational modes of adsorbed hydrogen is viable, the Free Energy change was determined for

complete desorption. The phonon DOS of the hydrogenated and pristine surface is shown in Figure

5(a) (the phonon dispersion relation is given in Figure S228). The most prominent changes occur

at 95 and 335 cm-1, where the phonon DOS is determined by Si-H vibrations. Some smaller

changes can be seen at 75 and 45 cm-1. Only the two larger contributions are taken into account

for this model. For a monoatomic solid, the Gibbs Energy within the Einstein model is given by36

, logħ

ħ . (9)

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FIGURE 5. (a) Phonon density of states (DOS) of hydrogenated and pristine Si surface. The

negative frequencies stem from the frozen bottom layers. (b) Free energy of complete desorption

for Einstein model, IP and AITD approach in comparison. The electronic energies were computed

with HSE06-D3.

A monoatomic solid has three degrees of freedom (i. e. phonon branches), thus a = 3 in

equation (9). The contribution of a single phonon branch for systems with more than one frequency

can be estimated by choosing a = 1 equation (9). For every adsorbed H atom, one phonon branch

at 335 cm-1 and two branches at 95 cm-1 show up. Thus, the Free Energy correction for the

hydrogenated surface is

335cm , 2 95cm , , (10)

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with N being the number of adsorbed H atoms per unit cell, here 8. The resulting Free Energy

curve is shown in Figure 5(b). While the temperature, at which desorption is predicted, is similar

to the AITD approach, it deviates from the more accurate IP approach and the curve is much more

flat. We conclude that the Einstein model is not accurate enough to describe the temperature

dependence of hydrogen coverage for the system at hand.

FIGURE 6. Pressure and temperature dependence of coverage for interpolated phonon approach

based on HSE06-D3 binding energies.

Finally, we investigated the pressure dependence of surface coverage p = 50 – 1013 mbar

as shown in Figure 6. Higher pressure shifts the coverage for a given temperature to higher values

as expected from Le Châtelier's principle. In agreement with the logarithmic dependence of Gibbs

energies on pressure,37 the effect of pressure change decreases with increasing total pressure. The

overall effect of pressure change is rather small and can thus be neglected in studies of MOVPE

reactions where the pressure changes are on the order of several tenth of mbar.

In comparison with the experimental results reported in Ref. [17], the data produced by the

most accurate computations (HSE06-D3 energies and EP corrections) give the best agreement of

the approaches tested here and a very good agreement in absolute terms. The experimentally

observed transition from pristine to hydrogen-covered surface under MOVPE conditions occurs

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around T = 1070 K.17 At this temperature, the IP method predicts a significant dehydrogenation

degree ( = 0.5 ML). Thus, the temperature where the pristine surface becomes predominant over

the hydrogenated phase is predicted consistently. The on-set of hydrogen desorption from the fully

covered surface occurs earlier in the theoretical description (T ≈ 700 K, Figure 4) compared to the

high-pressure experiments. But it is in very good agreement with previous TPD studies (700-780

K).15,16 In addition, the temperature range where desorption occurs seems to be narrower in

experiment compared to our results. For atmospheric pressure, the coverage at low temperatures

is underestimated, while it is overestimated at high temperatures, as is the case for MOVPE

pressure.

IV. CONCLUSIONS

The desorption of hydrogen from the fully hydrogenated H/Si(001) surface under

thermodynamic equilibrium conditions has been studied with a combination of density functional

theory, ab initio thermodynamics and phonon computations. In comparison to available

experimental data, the most accurate approach tested here (hybrid functional with dispersion

correction for the electronic energy and calculation of the phonon spectrum of the system) delivers

good agreement regarding the on-set temperature for desorption. Additionally, the temperature

range where desorption was observed experimentally is reproduced. However, the desorption

curve predicted here is less steep compared to experiment. It might be worthwhile to study the

persistence of residual hydrogen at high temperatures and the initial desorption at low temperatures

by experimental means. In the relevant temperature range for MOVPE, our computations predict

a significant ( < 0.95 ML) desorption starting at approximately T = 800 K. A nearly pristine

surface ( < 0.1 ML) is reached at approximately T = 1400 K. Beside the demanding computation

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of explicit phonons, the much more efficient interpolated phonon can be used in the future as well.

Ab initio thermodynamics with Einstein corrections are less accurate for the system at hand but

might be suitable for systems with heavier atoms.

This study now enables us to investigate adsorption and surface reactivity of organic and inorganic

precursor molecules relevant for thin-film growth with metal organic vapor phase epitaxy with an

accurate representation of the surface at the simulation temperature.

ACKNOWLEDGEMENTS

This work was supported by the Deutsche Forschungsgemeinschaft via Research Training Group

1782 “Functionalization of Semiconductors”. We thank Prof. Karsten Reuter (Munich) for his

hospitality during a research visit by P.R. and discussions. We thank Prof. Axel Groß (Ulm), Prof.

Wolfgang Stolz (Marburg) and Prof. Michael Dürr (Gießen) for discussions. Computational

resources were provided by the Hochschulrechenzentrum Marburg, the LOEWE-CSC Frankfurt

and the HLRS Stuttgart.

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Supporting Information for

Extend of hydrogen coverage of Si(001) under chemical vapor

deposition conditions from ab initio approaches

Phil Rosenow, Ralf Tonner

Table S1. D3-Parameters used for PBE-D3 and HSE06-D3 calculations (HSE-parameters are

those for PBE0). The expression for the dispersion correction is given according to S. Grimme,

S. Ehrlich, and L. Goerigk, J. Comp. Chem. 32, 1456 (2011) by :

with and .

The parameter is set to unity.

s8 a1 a2

PBE 0.7875 0.4289 4.4407

HSE06 1.2177 0.4145 4.8593

Table S2. Total energies of used unit cells in eV.

System PBE-D3 HSE06-D3

molecular H2 -6.74816899 -7.80958547

Si(001) -405.38169351 -469.43016295

= 0.25 ML -413.88348356 -479.30568130

= 0.50 ML -422.55333648 -489.35290438

= 0.75 ML -431.05702666 -499.23812543

= 1 ML (H/Si(001) -439.73108245 -509.29140990

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Figure S1. Phonon densities of states (DOS) of hydrogenated (top) and pristine (bottom)

surface cells with varying number of layers and supercell size for 6 layer cell. The vertical

dashed line in the top figure shows the on-set of Si-H vibrations (to higher vibrational energies).

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Figure S2. Phonon dispersion relation for the monohydrogenated (top) and pristine (bottom) surface. Branches from surface Si-H vibrations are colored in blue.

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Optimized coordinates of structures investigated in fractional coordinates (VASP

format)

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Extent of hydrogen coverage of Si(001) under chemical ... · thermodynamic equilibrium was studied...

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Transcript of Extent of hydrogen coverage of Si(001) under chemical ... · thermodynamic equilibrium was studied...